System for detecting peaks in vibrational energy spectra

ABSTRACT

The magnitude and frequency of a peak in detected vibrational energy emitted from a component that includes a periodic portion is provided to monitor the component for faults. A Fast Fourier Transform of the signal is taken to generate a spectrum. A maximum index in the spectrum is found between a determined frequency bound. An array of parameters is generated and a determined optimized transform root is used on the array of parameters. An interpolated peak location is estimated based on the array and an interpolated frequency is found based on the peak location. An interpolated magnitude is then determined. The use of the optimized transform root in the processing results in significant improvements to the magnitude and frequency estimations, which can improve, for example, detection of defects from vibration spectra of rotating components, estimates of energy radiated from an electronic component, or analysis of spectral content of ionizing radiation.

FIELD OF THE INVENTION

The present invention generally relates to systems and methods fordetecting and analyzing signals from components that emit vibrationalenergy. In particular, the present invention is directed to a system andmethod for determining the magnitude and frequency of a sinusoidalportion of a detected vibrational energy signal.

BACKGROUND

Common in many fields of science and industry is the estimation of themagnitude and frequency of measured or detected periodic, sinusoidalsignals generated by various components or systems. This estimationprocess is ubiquitous and the foundation of many standards andprocedures, including in aircraft component health monitoring, and ofteninvolves use of the Fourier transform. For example, in utility powergeneration and distribution systems, i.e., the electrical grid, it iscommon for line frequency to be run at 60 Hz. However, this frequency isactually 60 Hz +/−some small percentage error. In order to make sure thefrequency stays within acceptable margins, power providers mustdetermine both the frequency and magnitude of the voltage and thecurrent of electricity being generated at power plants. An accurate wayto determine these parameters involves performing a Fourier transform onthe measured data.

Similarly, the standards for both the International ElectrotechnicalCommission (IEC) and the Environmental Conditions and Test Proceduresfor Airborne Equipment (DO-160) require taking measurements ofelectromagnetic radiation related to the control of emission andtransmission of radiant energy from equipment. These measurements aretypically made using spectrum analyzers that incorporate Fouriertransform.

Likewise, in condition monitoring based on the detection of vibration,signals relating to rotating components of machines are monitored sothat mechanical diagnostics of the equipment may be performed while theequipment is in use. Periodic signals associated with any nonconformityof the rotating components, such as shaft imbalance, gear faults, andbearing faults, are predominantly quantified using spectral analysis.This spectral analysis can be done by measuring vibration with acalibrated accelerometer. The analog signal from the accelerometer isdigitized and an analysis is performed that uses Fourier transforms inorder to measure the magnitude of vibration at a given frequency. Thefrequency is associated with the properties of the rotating component.For example, a wind turbine may have a high-speed shaft driving agenerator at 1800 RPM (or 30 Hz). If the shaft has a 24-tooth pinion,the frequency associated with this gear is 720 Hz (24×30 Hz). The energyassociated with this frequency can be quantified using the Fouriertransform.

In another example, for the evaluation of the performance and load of aninduction motor, it is common to estimate slip. Slip is the percentdifference between the synchronous speed (i.e., line frequency) and thespeed of the motor. Excessive slip indicates the load on the motor isbeyond the design limit. It is usually assumed that synchronous speed is50 or 60 Hz, but at any given time the line frequency has error. Takingthe Fourier transform of the line current (using a current transformer),will give a more exact measure of the synchronous speed. And a moreexact determination of the motor speed can be made by measuring theshaft order 1 vibration with an accelerometer and then taking theFourier transform. Thus, estimating slip can be improved using theFourier transform.

In addition, many systems or tools that measure periodic signals need tobe calibrated to ensure accuracy. For example, it is common in the fieldof condition monitoring to test accelerometers with a handheld shaker(such as Model #394C06 available from PCB Piezotronics of Depew, NY)with an acceleration output of 1.00 G's (+/−3%) and an operatingfrequency of 159.2 Hz (+/−1%). In order to ensure compliance with anyspecification, it is important to accurately measure both the magnitudeand frequency of the shaker. This analysis may be validated using theFourier transform of the measured signal from the shaker.

Although Fourier transforms are useful in these applications, takingexact Fourier transforms can be impossible and is often impractical. Formost applications, then, a discrete Fourier transform (DFT) isimplemented. Because of the high computational load often required evenfor DFT, however, a Fast Fourier Transform (FFT), which is an algorithmthat computes the DFT of a sequence, or its inverse, is typically used.There is a need, however, in many applications, for improved FFT methodsthat can attain greater levels of accuracy for certain estimates, suchas for magnitude and frequency of measured periodic, sinusoidal signals,without significantly increasing computational load.

SUMMARY OF THE DISCLOSURE

A system for detecting faults for components of a helicopter includes aball bearing of a component of the helicopter, wherein the ball bearingincludes a cage, a ball, an inner race, and an outer race. Anaccelerometer is positioned to detect vibrational energy emitted fromthe ball bearing, wherein the vibrational energy includes vibrationalenergy emitted by the cage, the ball, the inner race, and the outerrace, wherein the vibrational energy includes at least one periodic,sinusoidal portion, and wherein the accelerometer generates a signalrepresentative of the detected vibrational energy. A processorconfigured to receive the signal and determine a frequency peak andmagnitude of the at least one periodic, sinusoidal portion based on anoptimized transform root.

In another embodiment, a system for estimating magnitude and frequencyof a signal representative of energy emitted from an apparatus includesa sensor configured to measure the signal representative of energyemitted from the apparatus, wherein the signal including a periodic,sinusoidal portion. A processor is configured to receive the signal andhas a set of instructions for taking a FFT of the signal using Welch'smethod to generate a spectrum, determining a maximum index in thespectrum between a determined frequency bound, generating an array ofparameters, the array of parameters being generated in part based onthree data points around the maximum index, determining an optimizedtransform root by minimizing an objective function that fits the arrayof parameters, determining an interpolated peak location of the spectrumbased on the array, determining an interpolated frequency of the peaklocation, and determining an interpolated magnitude of the spectrum atthe interpolated frequency.

In an embodiment, a method for estimating magnitude and frequency of asignal representative of energy emitted from a system includes sensingthe signal emitted from the system, wherein the signal includes aperiodic, sinusoidal portion. A spectrum is generated by taking a FFT ofthe signal, a maximum index in the spectrum between a determinedfrequency bound is determined, an array of parameters is determined, andan optimal transform root for estimating the magnitude and the frequencyof the periodic portion of the signal is determined. An interpolatedpeak location based on the array is determined, an interpolatedfrequency of the periodic portion of the signal based on the peaklocation is determined, and an interpolated magnitude of the periodicportion of the signal is determined.

In an embodiment, a method for estimating magnitude and frequency of asignal representative of energy emitted from a system includes measuringthe signal emitted from the system, wherein the signal includes aperiodic, sinusoidal portion, taking a FFT of the signal based on asample rate and a window length to generate a spectrum of the measuredenergy, determining a maximum index of the spectrum between a frequencybound, compiling an array of parameters, wherein the array of parametersis generated based on three points around the maximum index including afirst point from a maximum bin, a second point from a second bin justprior to the maximum bin, and a third point from a third bin just afterthe maximum bin, determining an optimal transform root, wherein theoptimal root and the parameters are used to find a parabolic functionrepresentative of the spectrum between the frequency bound, determiningan interpolated peak location of the spectrum within the frequency boundbased on the array of parameters and the parabolic function, determiningan interpolated frequency at the interpolated peak based on theinterpolated peak, the maximum index, the window length, and the samplerate, and determining an interpolated magnitude at the interpolated peakbased on the array of parameters and the interpolated peak location.

In addition, a system for detecting faults for components of ahelicopter is provided that includes a ball bearing of a component ofthe helicopter, wherein the ball bearing includes a cage, a ball, aninner race, and an outer race, an accelerometer positioned to detectvibrational energy emitted from the ball bearing, wherein thevibrational energy includes vibrational energy emitted by the cage, theball, the inner race, and the outer race, wherein the vibrational energyincludes at least one periodic, sinusoidal portion, and wherein theaccelerometer generates a signal representative of the detectedvibrational energy, and a processor configured to receive the signal anddetermine a frequency peak and magnitude of the at least one periodic,sinusoidal portion based on an optimized transform root. Further, theprocessor may be configured to execute a set of instructions to take aFFT of the signal using Welch's method to generate a vibrational energyspectrum, determine a maximum index in the vibrational energy spectrumbetween a frequency bound, generate an array of parameters, the array ofparameters being generated in part based on three data points locatednear the maximum index, determine an interpolated peak location of thevibrational energy spectrum using the optimized transform root,determine an interpolated frequency of the interpolated peak location,and determine an interpolated magnitude of the spectrum at theinterpolated frequency.

BRIEF DESCRIPTION OF THE DRAWINGS

For the purpose of illustrating the invention, the drawings show aspectsof one or more embodiments of the invention. However, it should beunderstood that the present invention is not limited to the precisearrangements and instrumentalities shown in the drawings, wherein:

FIG. 1 is a graph showing a raw FFT of a 159.2 Hz signal at 1 G using a4096 length FFT;

FIG. 2 is a graph of a parabolic interpolation of the spectrum shown inFIG. 1 ;

FIG. 3 is a graph showing bound on frequency error based a parabolainterpolation for raw FFT and transforms using linear, log, square root,cube root, fourth root, and fifth root transforms;

FIG. 4 is a zoomed view of a portion of the graph of FIG. 3 ;

FIG. 5 is a graph of an example of percent magnitude error forestimations based on raw FFT, linear, log, square root, cube root,fourth root, and fifth root transforms;

FIG. 6 is a zoomed view of a portion of the graph shown in FIG. 5 ;

FIG. 7 is a graph comparing percent frequency error of the fourth root,the fifth root, and the optimal root transforms;

FIG. 8 is a graph comparing percent magnitude error of the fourth root,the fifth root, and the optimal root transforms;

FIG. 9 outlines a process for estimating the magnitude and frequency ofa signal from a sinusoidal wave in accordance with an embodiment of thepresent invention; and

FIG. 10 a schematic view of a helicopter for which components such asball bearings can be monitored for faults in accordance with anembodiment of the present invention.

DESCRIPTION OF THE DISCLOSURE

For condition monitoring of gears and shafts based on the detection ofvibrational signals, there are a finite number of possible frequenciesbased on the parameters of the gear or shaft monitored because thevibrational frequencies of potential faults will be synchronous with therotational speed of the component. However, for some components, such asbearings and impact sensors, there are an infinite number of possiblefrequencies for potential faults because these types of components arenot synchronous with the rotation speed of an associated (i.e.,connected) rotating component. The vibrational signals from suchcomponents may include a sinusoidal wave and the present inventiondetermines the magnitude and frequency of a measured signal thatincludes a sinusoidal wave using a modified FFT, thereby improvingsignal resolution and thus fault detection for bearings and similarcomponents that are often part of systems with rotation components, suchas helicopters.

In the context of signal data analysis, the Fourier transform is used todecompose a time domain signal, which is the superposition offrequencies into the frequencies that make up the signal. Inimplementation, the Fourier transform is calculated using the FFTalgorithm. The FFT makes an assumption that the measured signal is afinite sequence of equally spaced measurements. The bandwidth, orNyquist frequency, of FFT, is half of the sampling rate (SR) of thediscrete signal. The frequency resolution of the FFT is then the SRdivided by the length, N, of the FFT, or SR/N.

It should be noted that there is some error in the FFT when thefrequencies are not exact integers of the sample rate. A key property ofsinusoids is that they are orthogonal at different frequencies; that is,A₁ sin(ω₁+ϕ₁) is orthogonal to A₂ sin(ω₂+ϕ₂). However, for sampledsinusoid signals having length N, such as those used in the FFT, thisproperty of orthogonality only holds for those harmonics that areinteger multiples of the sample rate divided by N. These frequencies,f_(k), can be expressed in terms of the sample rate, f_(s), as follows:

f _(k) =kf ₂ /N, where k=0,1,2, . . . , N−1   (Equation 1)

The FFT is defined only for the frequencies (ω_(k)) that are exactlyintegers of the sample rate (f_(s)), such as:

ω_(k)=2πkf _(s) /N   (Equation 2)

For frequencies that are not exact integers of the sample rate, it canbe shown that the FFT acts as a digital filter. The frequency responsefor any other frequency, ω_(k), is:

|X(ω_(x))|=|sin{(ω_(x)−ω_(k))NT/2}/sin{(ω_(x)−ω_(k))T/2}|  (Equation 3)

where X is the frequency response as a function of ω_(x), and T is theperiod of ω_(k). This results in spectral leakage and error in theestimation of the FFT amplitude for any given frequency, ω_(k), that isnot an integer of the sample rate. This spectral leakage is notminimized by increasing the length, N.

While there are a number of different algorithms for the FFT, many arebased on the Cooley-Tukey method, which is a sort of “divide andconquer” algorithm that recursively breaks the length of each data pointinto a composite of N=N₁×N₂, where N is the composite size and N₁ and N₂are smaller factors. The best known of these algorithms are based N₂=2,which results in a radix-2 algorithm. For these algorithms, the totallength (N) of the FFT is necessarily a 2^(n) power, for example, 4,096(2¹²) or 8,192 (2¹³).

One issue that sometimes needs to be addressed when using FFT for dataanalysis is the Gibbs phenomenon, which is observed in the Fourierseries of a piecewise periodic function that has a jump discontinuity ina signal and occurs with finite data lengths. A method used to controlthe Gibbs phenomenon and therefore obtain a better estimate the power ofa signal is Welch's method. Welch's method splits the signal into,typically, radix-2 lengths (the window length) and overlapped segments(the window length divided by two). The overlapped segments are thenwindowed, and a window function mitigates the Gibbs phenomenon. Atypical window function, applied to the segment, is the Hann window. TheHann window is smooth and bell shaped, and thereby mitigates theedge/jump discontinuity of a segment on which the FFT operates.

Welch's method returns the power of a signal in each frequency bin. Themagnitude of the signal is then the square root of the power of thesignal in a given frequency bin. The frequency resolution of each bin isa function of the sample rate and length to the window of Welch'smethod.

As an example, consider the calibration of a handheld shaker. Anacquisition system may use an accelerometer to measure voltage by ananalog-to-digital converter (ADC). The ADC sample rate is based on timesfixed by a clock or a crystal oscillator. Crystal oscillator devices,such as the NDK NX3225SA-12MHZ-STD-CSR-6 (made by NDK America, Inc.),have a frequency of 12 MHz +/−12 parts per million (ppm). The ADC samplerate is then the clock frequency divided by some radix-2 power. Forexample, a power of 11 would give an ADC sample rate of12,000,000/2,048, or 5,859 samples per second. Given the SR of 5,859samples per second and a Welch's window of 4,096, the resolution of eachFFT frequency bin is 1.4304 Hz.

If the vibration frequency of the handheld shaker is exactly 159.2 Hz,then the exact frequency bin on the FFT would be the 111.296 bin index(i.e., 159.2/1.4304). Unfortunately, the resolution of the FFT, aspreviously shown, is finite, therefore some of the energy in the 159.2Hz acceleration is found in bin 111 (corresponding to a frequency of158.776 Hz), and some is found in bin 112 (corresponding to a frequencyof 160.21 Hz), as can be seen in graph 100 in FIG. 1 , which shows themagnitude as a function of frequency of spectrum 104. This is an exampleof spectral leakage. Note also in FIG. 1 that at the maximum magnitudeis 0.944 G's, which translates to a 5% error for the measured inputsignal for the 1 G handheld shaker. In this example, an error of thatamount would mean the handheld shaker would fail the calibration.

In general, a strategy for improving the calibration process may be totry to increase the resolution by either reducing the sample rate orincreasing the window length of the FFT. If the window in the aboveexample was increased to 32,768, then the bin index would become 890(159.2/(5,859/32,768)). This corresponds with a frequency of 159.13,which is closer to true frequency, but the magnitude is then only 0.915G's because more energy is found in bin index 891. At noted earlier,increasing N does not reduce spectral leakage.

In the present invention, in order to better estimate the magnitude andfrequency of a measured signal, the estimation of magnitude andfrequency incorporates an interpolation technique into the Fouriertransform. As can be seen in spectrum 104 of the example shown in FIG. 1, the maximum energy is likely found at a frequency between 158.8 Hz and160.2 Hz. Further, there is a not insignificant amount of energy at157.3 Hz. Therefore, interpolation may allow the maximum energy valueand the frequency at which it occurs to be estimated by assuming acontinuous function or distribution. A polynomial quadratic, which is asmooth function, can be used to interpolate the maximum energy through,for example, a triplet of points.

A polynomial quadratic has a general formula that can be expressed as:

y(x)=a(x−p)² +b   (Equation 4)

where a is the curvature and depends in this context on the window usedin the FFT; p is the center point and gives the interpolated peaklocation; and b is the amplitude that here equals the peak amplitude ofthe FFT spectrum. The three sample points nearest the peak can bedefined as:

y(−1)=α, y(0)=β, and y(1)=λ,

where the bins about those three points are defined at [−1 0 1].Substituting the bin values for x in Equation 4 and simplifying resultsin the following expressions for α, β, and λ:

α=ap ²+2ap+a+b   (Equation 5)

β=ap ² +b   (Equation 6)

λ=ap ²−2ap+a+b   (Equation 7)

By combining equations, substituting, and rearranging terms, thefollowing relations can be derived:

α−λ=4ap   (Equation 8)

p=α−λ/4a   (Equation 9)

α=ap ²+(α−λ/2)+a+(β−ap ²)   (Equation 10)

a=½(α−2β+λ)   (Equation 11)

Then, the interpolated peak location, p, in bins, can be expressed as:

$\begin{matrix}{p = {1/2\frac{\alpha - \lambda}{\alpha - {2\beta} + \lambda}}} & \left( {{Equation}12} \right)\end{matrix}$

If k is the index of the maximum spectral value, then the interpolatedfrequency (frq) would be determined as follows:

frq=(k+p)SR/window length   (Equation 13)

The interpolated magnitude (mag) could be determined from the following:

mag=y(x)=β−¼(α−λ)p   (Equation 14)

In the example data from FIG. 1 , the three points at 157.346 Hz,158.776 Hz, and 160.21 Hz, having energies of 0.2895 G, 0.9447 G, and0.7184 G, respectively, can be used to interpolate the maximum energy.As can be seen from graph 108 shown in FIG. 2 , these three points serveas the points through which a polynomial 116 is fitted for the data forspectrum 104, and are used to find values for α, β, and λ for the data.With no further transformation of this data, the above interpolationtechnique results in a peak value of 0.9708 G with a frequency of159.1246 Hz.

These estimations of the magnitude and frequency are based on theinterpolation without any additional transform. However, a Gaussianwindow magnitude is precisely a parabola on a dB scale, which impliesthat use of the above quadratic interpolation technique is exact dBscale. If, therefore, there is a transformation (perhaps logarithmic,which gives dB scale, or some other transform) that defines therelationship of three points in the spectrum (here, α, β, and λ) inwhich β is the maximum bin, α is the bin just prior the maximum, and λis the bin just after the maximum, such that the three points lie on aparabola, then such a transformation would minimize the error betweenthe true magnitude and frequency and the interpolated magnitude andfrequency of the measured signal. A log transform is one option,although a log transformation tends to compress the peak values. Otherpotential transformations include the square root, the cube root, thefourth root, and the fifth root.

A test for such a transform would be bounded by the minimum errorbetween the points of the interpolation and those points tested by thetransform. The minimum error would then occur where a FFT bin is exactlythe frequency of the sinusoidal signal being processed. For the givenexample with a SR of 5,859, a window of 4,096, and a desired frequencyof 159.2, the exact frequency of zero error (at bin 111) would be158.7766 Hz (111*5,859/4,096). The bound from bin 111 to bin 112 wouldbe a frequency of 160.2070 Hz (112*5,859/4,095). The maximum magnitudeerror would then occur at 111.5, or a frequency of 159.4918 Hz.

At a high level, the test involves first taking the FFT of the measuredsinusoidal signal for a frequency between a bound, in this examplebetween 158.7766 Hz and 160.2070 Hz, using Welch's method. The spectrumdata is then transformed by taking the log, the square root, or one ofthe other proposed transforms. Then the bin with the maximum value(i.e., the bin containing β) is determined and is used to find a valueof k. The value of k can be used to calculate p, and then the associatedinterpolated frequency and transformed magnitude can be found. Takingthe inverse transform of the calculated transformed magnitude allows theerror between the interpolated magnitude and the magnitude of thesinusoidal signal to be evaluated.

FIG. 3 shows a graph 120 of the frequency determined by interpolation asdescribed above with respect to a measured signal and the associatedpercentage error for those frequencies. Results are shown for raw FFTdata 121 and for estimates made via the parabolic interpolation based ona linear transform 122, a log transform 123, a square root transform124, a cube root transform 125, a fourth root transform 126, and a fifthroot transform 127. The raw FFT results 121 show greater error comparedto any of the parabolic estimates (122-127). In FIG. 4 , a zoomed viewof a portion of graph 120 is shown around 159 Hz, where it can be moreeasily seen that the estimates based on fourth root transform 126 andfifth root transform 127 show the least error.

FIG. 5 is a graph 128 from a measured signal processed as describedabove of input frequency and the associated magnitude percentage errorfor a raw FFT 129, and for parabolic estimates based on a lineartransform 130, a log transform 131, a square root transform 132, a cuberoot transform 133, a fourth root transform 134, and a fifth roottransform 135. A maximum error of 14% in raw FFT 129 is large in thiscontext. In FIG. 6 , a zoomed view of a portion of graph 128 is shown,where it can be more easily seen that the estimates based on the fourthroot transform 134 and the fifth root transform 135 show the leasterror.

As can be seen, different transforms result in different percent errorsfor a given sinusoidal signal. An ideal root for minimizing percenterror for a given signal can be estimated via an optimization approach.The optimization would require an objective function, which for thistype of situation could be the mean square error, for example. That is,over the frequency in interest, for each trial transform, theperformance metric will be the mean square error over the boundedfrequencies. An optimization technique, such as Newton-Raphson method orother root-finding algorithm, can be used to find the optimal root. Inthe given example, the optimal root is determined to be the 4.3093^(th)root.

The optimal root is determined by finding the root that minimizes thepercent error for an idealized function that produces a signal with anamplitude of one. Since this approach is based on minimizing percenterror, it is generalized. For example, a system may have a sample rateof 5,859 samples per second. For four seconds of data, the spectrum (FFTwindow in Welch's method) is 4,096 with an overlap of 2,048. The signalof interest is, for example, 159.2 Hz. From DFT theory, each FFT bin isthe sample rate divided by the window, which here is 1.403 Hz. Ideally,the exact bin for the frequency would be 159.2/1.403 or bin 111.296.However, since the bin is an integer, the frequency will fall betweenbin 111 and bin 112.

To solve and find the best transform, a sample of idealized frequenciesis complied, from bin 111 (111*1.403=158.7766 Hz) to 112 (160.2070 Hz).The space is partitioned 50 times, which creates an array with a lengthof 23,436. Since each bin to bin+1 is the same size, the accuracy of anFFT can be tested for a known transform with a known signal source, s,which has an amplitude of one. Thus, the magnitude of any trail (i.e.,the output of the FFT) should be 1, and the frequency of the signalsource is denoted as crtfrq. The following pseudo code determines s andcrtfrq:

-   -   step=sr/win/50    -   idx=1, 2, 3, . . . 5859×4    -   for i=1:50    -   crtfrq=158.7766+step*i;    -   s=sin(idx/sr*2*pi*frq).

Then the optimal root can be determined by minimizing the percent errorin the amplitude, which can be accomplished as follows:

-   -   j=30;    -   sr=5859;    -   n=5859*4;    -   win=4096;    -   df=sr/win;    -   idx=159.2/df;    -   % idx=round(159.2/df);    -   % % idx=216;    -   % fq=df*(idx−0.50);    -   start=df*floor(idx);    -   en=df*round(idx+1);    -   frq=linspace(start,en,j);    -   scale=1;    -   for i=1:j    -   s=sin((1:n)/sr*2*pi*frq(i))*scale;    -   [mg(i),fq(i)]=quadInterSpecPeakOps(s,win,root);    -   end    -   me1=−(scale−mg)/scale*100;    -   mse=mean(me1·{circumflex over ( )}2);    -   end    -   function [mag,freq]=quadInterSpecPeakOps(v,win, root)    -   %[mag,freq]=quadInterSpecPeak(v,sr)    -   %v: is the signal    -   %sr: is the sample rate    -   %quadratic interpolation of the spectral peaks    -   sr=5859;    -   [spc,˜]=psde(v,win,sr,win/2);    -   [˜,idx]=max(spc);    -   abg=spc([idx−1 idx idx+1]);    -   abg=nthroot(abg,root);    -   p=½*(abg(1)−abg(3))/(abg(1)−2*abg(2)+abg(3)); %interp peak bin    -   freq=(idx+p−1)*sr/win;    -   mag=abg(2)−¼*(abg(1)−abg(3))*p;    -   mag=mag·{circumflex over ( )}root;    -   end

In FIG. 7 , a graph 136 is shown of the frequency error for themagnitude estimations based on a fourth root transform 137, a fifth roottransform 138, and a transform 139 based on the determined optimal rootof 4.3093. In FIG. 8 , a graph 144 shows the percent error for magnitudeestimates based on a fourth root transform 145, a fifth root transform146, and a transform 147 based on the determined optimal root of 4.3093.The optimal root of 4.3093 was robust in that it did not change withchanges in window size, such as from 8,192 to 16,384 to 32,768.

Table 1 compiles the maximum percent magnitude error, maximum percentfrequency error, and mean magnitude error for estimations in the aboveexample based on the various transformations that can be used as part ofthe interpolation. Transform Max. Mag. Error Max. Freq. Error Mean Mag.Error RAW FFT 14.16% 0.433%  −4.99% Linear 6.37% 0.047%  −2.48% Log (dB)3.58% 0.014%    1.27% 4^(th) Root 0.18% 0.001%  −0.09% 5^(th) Root 0.48%0.002%    0.15% 4.3093^(th) Root 0.06%  0.0003% −0.009%

As can be seen from the values in Table 1, the magnitude estimationdetermined using the optimal root transform (in this example, the4.3093^(th) root) derived as described above is on average 190 timeslower in error compared to the log transform. For the frequencyestimation, the result based on the present invention has 60 times lowerpercent error than the log transform. Using the optimal root for thetransform thus greatly improves the mean and maximum error of theparabola interpolation. This in turn improves the estimated values ofspectral analysis of signals processed using a FFT based on the optimalroot transform.

Turning to FIG. 9 , an overview of an estimation process, such asestimation process 200, is shown that includes measuring a periodicsignal at step 204 and inputting a sample rate and a window length atstep 208. At step 212, a FFT is taken using Welch's method, whichgenerates a spectrum 216 for the measured signal. Then, at step 220, themaximum index in spectrum 216 is found between a determined frequencybound. At step 224, an array is compiled of the α, β, and λ parametersfrom three points that serve as the points through which a polynomial isfitted for the data for spectrum 216. The optimal transform root of4.3093, as determined as described above, is used at step 228 to on thearray of parameters, which minimizes the percent error of the estimatedfrequency and magnitude. From those parameters, an interpolated peaklocation, p, is determined at step 232. Then, at step 236, theinterpolated frequency is determined using Equation 13, and at step 240the interpolated magnitude is determined using Equation 14.

In an embodiment, a system for monitoring ball bearings in a componenton a helicopter is provided in which a vibrational spectrum couldinclude a frequency of interest at any frequency. An accelerometer ispositioned in proximity of the ball bearing such that vibrational energyfrom the bearing is detected. Frequency peaks associated with potentialfaults are determined as described above using an optimized transformroot, and the faults may be associated with the inner race, the outerrace, the cage, or the ball of the monitored bearing.

In FIG. 10 , there is shown a helicopter 300, although it will beunderstood that the vibrational spectral analyses described herein areapplicable to components is various types of vehicles and equipment. Ata high level, helicopter 300 includes a rotor system 304, a plurality ofblades 308, a fuselage 312, a landing gear 316, and a tail assembly 320.Rotor system 304 is generally designed and configured to rotate theplurality of blades 308 and can include a control system for selectivelycontrolling the pitch of each the plurality of blades 308 so as tocontrol direction, thrust, and lift of helicopter 300. Rotor system 304is coupled to fuselage 312, which is generally designed to carry theoperator and other passengers, among other things. Fuselage 312 iscoupled to landing gear 316, which supports helicopter 300 whenhelicopter 300 is landing and/or when helicopter 300 is at rest on theground. Tail assembly 320 represents the tail section of the aircraftand can feature component similar to rotor system 304 and includesblades 324. Blades 324 may provide thrust in the same direction as therotation of blades 308 so as to counter the torque effect created byrotor system 304. Many of the systems on helicopter 300 include variouscomponents that undergo wear, tear and fatigue damage over time, and somay exhibit faults embedded in vibrational energy emissions. For anysuch components that may include faults that could be found at anyfrequency, a sensor, such as an accelerometer, may be positionedproximate to the component in order to detect vibrational energy emittedby the component. Further, the spectral analysis performed on thedetected vibrational energy may be done on the accelerometer itself oranother component on helicopter 300 since the analysis described hereindoes not require a large computational load.

Because the improved estimation of magnitude and frequency does notrequire significant additional computational load, it can be integratedinto a number of applications, including determining the energyassociated with a bearing cage, ball, inner or outer race defect from avibration spectrum. In addition, it may be used to estimate energyradiated from a microcontroller or other electronic device for thepurpose of determining whether there will be electromagneticinterference. Another application would be for estimating the soundpressure level from a transformer or other device, or for analyzing thespectral content of ionizing radiation.

Exemplary embodiments have been disclosed above and illustrated in theaccompanying drawings. It will be understood by those skilled in the artthat various changes, omissions, and additions may be made to that whichis specifically disclosed herein without departing from the spirit andscope of the present invention.

What is claimed is:
 1. A system for detecting faults for components of ahelicopter comprising: a ball bearing of a component of the helicopter,wherein the ball bearing includes a cage, a ball, an inner race, and anouter race; an accelerometer positioned to detect vibrational energyemitted from the ball bearing, wherein the vibrational energy includesvibrational energy emitted by the cage, the ball, the inner race, andthe outer race, wherein the vibrational energy includes at least oneperiodic, sinusoidal portion, and wherein the accelerometer generates asignal representative of the detected vibrational energy; and aprocessor configured to receive the signal and determine a frequencypeak and magnitude of the at least one periodic, sinusoidal portionbased on an optimized transform root.
 2. A system for detecting faultsfor components of a helicopter comprising: a bearing of a component ofthe helicopter, wherein the bearing includes a cage, a ball, an innerrace, and an outer race; an accelerometer positioned to detectvibrational energy emitted from the bearing, wherein the vibrationalenergy includes vibrational energy emitted by the cage, the ball, theinner race, and the outer race, wherein the vibrational energy includesat least one periodic portion, and wherein the accelerometer generates asignal representative of the detected vibrational energy; and aprocessor configured to receive the signal and determine a frequencypeak and magnitude of the at least one periodic portion based on thesignal and an optimized transform root.